Omega Transmission Lines with Applications to Effective Medium Models of Metamaterials

0.84 MB·

by  Joni Vehmas

Omega transmission lines with applications to effective medium
models of metamaterials
J. Vehmas,1 S. Hrabar,2 and S. Tretyakov1
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Department of Radio Science and Engineering/SMARAD Center of Excellence,
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Aalto University, P. O. Box 13000, FI-00076 Aalto, Finland.
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Email: joni.vehmas@aalto.fi
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r 2 Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3,
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10000 Zagreb, Croatia.

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In this paper we introduce the concept of transmission lines with inherent bi-

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h anisotropyandestablish ananalogybetween these linesandvolumetric bi-anisotropic
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- materials. In particular, we find under what conditions a periodically loaded trans-
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a mission line can be treated as an effective omega medium. Two example circuits are
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introduced and analyzed. The results have two-fold implications: opening a route to
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emulateelectromagneticpropertiesofbi-anisotropicomegamediausingtransmission-
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line meshes and understanding and improving effective medium models of composite
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materials with the use of effective circuit models of unit cells.
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I. Introduction
Omega media, introduced in Refs. 1 and 2, is a widely researched reciprocal special
case of bi-anisotropic media. In bi-anisotropic media, electric field induces both electric and
magnetic polarizations (and the same is true for magnetic field). The omega material is
reciprocal and it is characterized by an antisymmetric magnetoelectric coupling dyadic. In
case of the uniaxial omega media2 the material relations can be written as
D = ǫ E+jK√ǫ µ z H
0 0 0
· × (1)
B = µ H+jK√ǫ µ z E.
0 0 0
· ×
Here, EandHareelectricandmagneticfieldswhileDandBstandforelectric andmagnetic
flux densities, respectively. Furthermore, K is the omega coupling coefficient, ǫ and µ are,
0 0
respectively, the vacuum permittivity and permeability, and the unit vector z defines the
0
only preferred direction in the uniaxial sample. The permittivity and permeability dyadics
(denoted as ǫ and µ) are symmetric and uniaxial.
Omega materials have aninteresting property of having different wave impedances for op-
posite propagation directions. This implies asymmetry of reflection coefficients at interfaces
with omega media when the illumination direction is reversed. Furthermore, the matching
condition for interfaces between omega media and conventional magnetodielectric media de-
pends on all three material parameters, opening interesting design possibilities in antenna
and microwave engineering2,3. Most of recently introduced terahertz and optical metamate-
rialstructureshavethesymmetrycorrespondingtotheomegatypeofbi-anisotropiccoupling
(for example, arrays of complex-shaped metal particles or meshes positioned on one side of
a dielectric substrate or similar multi-layer structures). Understanding and modeling of
magnetoelectric omega coupling is a pre-requisite for understanding of effective response of
these advanced electromagnetic materials and developing new applications.
It has been suggested that omega media can be realized by embedding electrically small
resonant metal particles of an appropriate shape (e.g., Ω-shaped metal inclusions, that
is, centrally connected small dipole and loop antennas), into a conventional dielectric1.
However, as such wire omega particles are resonant structures with all the polarizabilities
(electric, magnetic, magneto-electric, and electro-magnetic) resonant always at the same
frequency, the operational bandwidth, i.e., the bandwidth where K is significantly different
from zero, is rather narrow. This also means that the tunability of medium parameters
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in general is very limited and such media always have band-stop behavior. Losses in such
media are typically also quite high when K is significantly large. For these reasons, the
possibilities for practical applications are limited. Here, we consider possibilities of realizing
omega media with periodically loaded transmission lines (TLs). First, we compare the wave
impedanceofomegamediawiththeBlochimpedanceofageneralperiodicallyloadedTLand
derive the required conditions for omega-like response. Second, a T-type circuit topology
is considered to fulfill the required conditions. Third, a circuit topology using coupled
inductors is analyzed. It should be emphasized that the goal, here, is not to replicate the
narrow-band dispersion of any of the proposed realizations based on resonant particles, but
rather to study more general media that still satisfy the material relations for omega media.
As will be shown below, the omega coupling parameter is defined by the asymmetry of
unit cells of periodical structures (transmission lines in this example). This has important
implications for problems of homogenization of composite materials (metamaterials) and
electrically thin composite layers (metasurfaces). The unit cells (or periodically repeated
planes of various inclusions) can be modeled by equivalent T- or Π-circuits, arranged in a
periodicalfashion. Herewe willshow thattheasymmetry ofthese unitcells (mostcommonly
imposed by the topology of the sample interfaces) can be properly accounted for by the
effective omega-coupling parameter. This can remove the common problem of non-physical
anti-resonance in effective permittivity and permeability, extracted from the reflection and
transmissioncoefficients ofplanarslabs, withouttheneedtointroduceadditionalparameters
which explicitly depend on the propagation constants of partial plane waves in the medium
(as in Refs. 4 and 5). From the physical point of view, the effective omega parameter
accounts for first-order spatial dispersion effects in materials with non-negligible electrical
size of the unit cells.
II. Omega media and omega transmission lines
A. Propagation constant and wave impedance
The propagation constant for axially propagating plane waves in omega media can be
easily derived from (1) and is given by2,3
β = k ǫ µ K2 = k √ǫ µ 1 K2, (2)
0 t t − 0 t t − n
q q
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where k is the free-space wave number, ǫ and µ are, respectively, the relative transverse
0 t t
permittivity and permeability, and K is the normalized omega coefficient defined as K =
n n
K/√ǫ µ . For lossless media with ǫ µ > 0, K is purely real. Therefore, for such media the
t t t t n
propagation constant is real, i.e., there is wave propagation only when we have K < 1.
n
| |
An interesting property that separates omega media from conventional magnetodielectric
media is that the wave impedance is different for waves traveling in the opposite directions.
The wave impedance for the axial propagation can be written as2,3
µ µ
Z = 0 t 1 K2 jK , (3)
Ω s ǫ0ǫt − n ± n
(cid:16)q (cid:17)
where the two solutions correspond to opposite axial propagation directions.
B. Required conditions for unit cells
Our goal will be to emulate the wave-propagation properties of omega media with peri-
odically loaded transmission lines. Bloch impedance can be considered as the characteristic
impedance of periodically loaded transmission lines. It is defined simply as the ratio of the
voltage and current at the terminals of the unit cell. It should be noted that the value of
the Bloch impedance depends on how the terminal points are chosen and is, therefore, not
unique for a given unit cell. The Bloch impedance for a general reciprocal periodic structure
is defined using ABCD-parameters of unit cells as6
2B
Z = . (4)
B
∓A D (A+D)2 4
− ∓ −
q
Here, the two signs correspond to different propagation directions and the current is defined
to flow always in the direction of the energy propagation. It should be noted that the
top sign does not necessarily always lead to the correct solution for the positively traveling
wave and the bottom sign for the negatively traveling wave, but the solutions may switch.
Incorrect choice of sign leads to the non-physical result of negative real part of the Bloch
impedance for passive structures. Taking this into account, the Bloch impedance for waves
traveling along the positive (+) and negative ( ) directions can be written as
−
jB A+D 2 D A
ZB± = 1 j − . (5)
AD 1 s − 2 ± 2 
− (cid:18) (cid:19)
 
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Comparing the two impedances of (5) to the wave impedances of omega media (3) and
assuming them to be equal, we can write the normalized omega coefficient K in terms of
n
the Bloch impedances ZB+ and ZB−:
ZB+−ZB−
K = ZB++ZB− . (6)
n
1 ZB+−ZB− 2
− ZB++ZB−
r
(cid:16) (cid:17)
This can be further written using the ABCD parameters as
D A 1
K = − . (7)
n
2 √1 AD
−
Therefore, as long as we have A = D, i.e, the unit cell is asymmetric, and AD is finite,
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the normalized omega coefficient K is non-zero. Furthermore, we can also determine the
n
effective magnetodielectric wave impedance µ µ /(ǫ ǫ ) based on (3) and (5). This can be
t 0 t 0
q
written as
µ µ jB
0 t
= . (8)
s ǫ0ǫt −√1 AD
−
Knowing the effective normalized omega coefficient and the effective wave impedance, we
can also extract the effective refractive index by comparing the dispersion in omega media
(2) and the dispersion in the periodically loaded TL. The latter can be calculated easily for
any unit cell using basic ABCD-matrix theory and the Floquet theorem and is given by6
j A+D (A+D)2 4(AD BC)
β± = ln ± − − , (9)
∓d  q 2 
 
where d is the period of the structure. For reciprocal unit cells (AD BC = 1), the two
−
solutions are equal and (9) simplifies to
j A+D A+D 2
β = ln + 1 . (10)
−d  2 s 2 − 
(cid:18) (cid:19)
 
Knowing the effective normalized omega coefficient, the effective refractive index of the TL
can be determined by comparing (10) to (2). Moreover, knowing the effective refractive
index and wave impedance we can easily find the effective permittivity, permeability, and
the (denormalized) omega coefficient.
C. T circuit
The simplest possible TL loading element is a T-type circuit (or alternatively Π-type), as
we need an asymmetric circuit for the Bloch impedances for different propagation directions
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to be different. Let us take a look at a TL periodically loaded with T-type circuits as shown
in Fig. 1. Let us also assume that the period of the unit cell d is very small electrically. In
this case, the ABCD-parameters, calculated by simply multiplying the ABCD matrices of
each element in the unit cell in the right order, have the form
FIG. 1: The unit cell under study.
A = 1+Y Z , B = Z +Z +Y Z Z , C = Y , D = 1+Y Z . (11)
2 1 1 3 2 1 3 2 2 3
Therefore, the normalized omega coefficient, as defined in (7), can be written as
Y (Z Z )
2 3 1
K = − (12)
n
2 1 (1+Y Z )(1+Y Z )
2 1 2 3
−
q
and the magnetodielectric wave impedance as
µ µ j(Z +Z +Y Z Z )
0 t 1 3 2 1 3
= . (13)
s ǫ0ǫt − 1 (1+Y2Z1)(1+Y2Z3)
−
q
Again, we can see that for a symmetrical unit cell (Z = Z ) we have K = 0.
1 3 n
Let us further simplify the analysis by assuming that we have Z = 0 Ω, that is, the
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structure corresponds to the cascade shown in Fig. 2a. In this case, we have simply
√ Y Z µ µ Z
2 1 0 t 1
K = − , = . (14)
n
2 s ǫ0ǫt sY2
From (14), rather surprisingly, it can be seen that even a TL periodically loaded with a
simple two-port consisting of a series inductor and a shunt capacitor can be interpreted
as an omega media with K = ω√LC/2 and µ µ /(ǫ ǫ ) = L/C. Notably, if we would
n 0 t 0 t
q q
have chosen Z to be zero instead of Z , the magnetodielectric wave impedance would be the
1 3
same but the normalized omega coefficient would have a different sign (K = ω√LC/2).
n
−
However, itiswellknownthattheequivalent circuitofaninfinitesimal sectionofanunloaded
infinite TL can also be considered as a two-port consisting of a series inductor and a shunt
capacitor. Therefore, loading a regular TL with these lumped elements is typically believed
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to just increase the equivalent permittivity and permeability with no need for any extra
effective parameters. In fact, if we define the terminal points of the unit cell so that the unit
cell is symmetric (i.e., series-shunt-series loading with the element values Z/2, Y, and Z/2
as shown in Fig. 2b), we have D = A and thus, according to (7), K = 0 at all frequencies.
n
As we consider here aninfinite cascade, both unit cell choices illustrated in Fig. 2 are equally
valid. Obviously, the dispersion in both cases is the same as the physical structure is the
same (assuming that d is electrically small). On the other hand, looking at (2) the different
values of K in these two cases indicate that the effective permittivity and permeability
n
should be defined differently! In the former case, the stopband is interpreted to appear due
to having K > 1 whereas in the latter case it appears due to the term √ǫ µ . Let us look
n t t
| |
at an example in order to clarify this confusion.
FIG. 2: By defining the terminal points in different ways, the unit cell in an infinite
cascade can be asymmetric (a) or symmetric (b). Though dispersion is the same for the
two structures, the Bloch impedance and therefore also the effective material parameters
are different.
Let us first consider the asymmetric unit cell shown in Fig. 2a with Z = jωL and
Y = jωC having the component values L = 20 nH and C = 1 pF. We also add some small
losses in the series branch (R = 0.001 Ω). This is done just to avoid numerical problems
in calculations and does not change the general conclusions. The period of the structure
is 2 mm. The effect of the electrically short TL segments is neglected, as before. Let us
first take a look at the Bloch impedance and dispersion in an infinite cascade of such unit
cells shown in Figs. 3 and 4. As can be expected, the circuit has low-pass behavior. In
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the passband, the Bloch impedance is complex. The real part of the Bloch impedance is
the same for both propagation directions while the imaginary part has the same magnitude
but different signs. In the stopband, the real part of the Bloch impedance is zero for both
propagation direction, but the imaginary part has different signs and magnitudes. Notably,
the wave impedance term corresponding to µ/ǫ in omega media (denoted with the black
q
line in Fig. 3) is constant from 0 to 4 GHz. The normalized omega coefficient K is shown in
n
Fig. 5. As can be expected, as K becomes larger than one, there is no propagation. Finally,
n
in Fig. 6 the permittivity, permeability, and the omega coefficient K = K √ǫ µ are shown.
n t t
In the passband, permittivity, permeability, and omega coefficient are real and positive and
their values increase as we get closer to the stop band. They show resonant behavior in
between the passband and the stopband and have negative values in the stopband.
Now, let us look at the symmetric unit cell of Fig. 2b with Z = jωL/2 and Y = jωC
having the component values L = 20 nH and C = 1 pF. As before, the period is 2 mm and
we have included some small losses in each series branch (R/2=0.0005 Ω). As the period is
electrically small and the effect of the TL segments can therefore be neglected, this physi-
cally corresponds to the earlier case when an infinite cascade is considered. Obviously, this
means that the dispersion is the same as before. The Bloch impedance, on the other hand,
is the same for both propagation directions in this case. The Bloch impedance is equal to
the magnetodielectric wave impedance of (13) and can be written as Z = L/C ω2L2/4.
B
−
q
Therefore, the Bloch impedance is constant with the value Z = L/C only if the product
B
q
ω2L2/4 is sufficiently small. Clearly, if we consider a segment of a conventional, truly ho-
mogeneous TL with distributed series inductance and shunt capacitance, the term ω2L2/4
always approaches zero as the length of the segment approaches zero. Also, if we extract
the material parameters, the permittivity and permeability are notably different than in
the previous case, as can be observed from Fig. 7. Instead of both permittivity and per-
meability increasing in the passband, the permeability decreases. Typically, when we want
to homogenize a periodical structure, we look at the lower passband where the permittivity
and permeability are weakly dispersive. In this frequency range, the two solutions are ap-
proximately equal, as K is much smaller than unity in the asymmetric case. In fact, the
n
| |
negative frequency derivative of the permeability is not a physical result for passive media
as for passive low-loss media we must have ∂ǫ(ω) > 0 and ∂µ(ω) > 0 7. Also, the imaginary
∂ω ∂ω
part of the permeability is positive in the stopband which also implies that the medium
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appears to be active. Therefore, this parameter extraction can only be valid at very low
frequencies where the derivatives are approximately zero. This is not true for the parameter
extraction in the previous asymmetric case, as in that case both derivatives are positive at
all frequencies, and the anti-resonant artifact of one of the material parameters is removed.
Notably, this only happens when the inductance of the other series inductor in the T-circuit
is reduced to zero. Even if asymmetry (i.e., omega coupling) is introduced into the full T-
circuit by making the inductance of one of the series inductors smaller than the inductance
of the other one, the problem of non-physical permittivity and permeability remains as long
as both inductors have non-zero values.
For an infinite structure, there is no physical reason to select one unit cell topology
over the other. However, as soon as we consider finite-sized samples, we will see that the
topologies of the first and last unit cells in fact define the symmetry or asymmetry of the
unit cell and in this way also uniquely define the omega coupling coefficient. If the overall
sample is formed by a set of n complete unit cells, the effective parameter model with three
effective parameters becomes unique. If at one of the interfaces the unit cell is incomplete,
this can be accounted for by an addition of a series or parallel circuit element (equivalent
to an additional surface current sheet in the effective medium model) or alternatively the
sample can be interpreted as a set of n 1 complete unit cells terminated in an extra circuit
−
element. Moreover, a set of n complete asymmetric unit cells can be transformed into n+1
complete symmetric unit cells by adding a series element and a parallel element to one end
of the cascade and a single series element to the other. This is illustrated in Fig. 8.
III. Omega parameter in effective medium models of metamaterials and
metasurfaces
Periodically loaded transmission-line model or a periodical chain of connected two-port
networks is one of the common approaches in modeling composite materials. Here, the elec-
tromagnetic properties of each period are modeled by the S- or Z-parameters of a two-port
network, connected by short sections of transmission lines or directly. The main simpli-
fication in this model is that the adjacent layers of a multi-layer structure interact only
by the single fundamental plane-wave harmonic, which is valid when the period along the
propagation direction is larger than the period in the transverse plane.
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500
real(ZB+)
400
imag(ZB+)
real(ZB−)
300
imag(ZB−)
Ω] 200 real wave impedance
[
B
Z 100
0
−100
−200
0 1 2 3 4
frequency [GHz]
FIG. 3: Bloch impedance for the asymmetric unit cell of Fig. 2a with Z = jωL and
Y = jωC having component values L = 20 nH and C = 1 pF.
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real(β)
3.5
imag(β)
’”light line”
3
z]
H2.5
G
[
cy 2
n
e
u
q1.5
e
r
f
1
0.5
0
−1500 −1000 −500 0 500 1000 1500 2000
propagation constant [1/m]
FIG. 4: Dispersion in an infinite cascade for both unit cells shown in Fig. 2 with Z = jωL
and Y = jωC having component values L = 20 nH and C = 1 pF.
For a spatially infinite periodical material a T- or Π-circuit can be equally adopted for
anypassive structure. Ifthestructure topologyissymmetric withrespect to thepropagation
direction (the simplest example is a 3D periodical array of spherical inclusions), it is natural
toadoptasymmetrical T-orΠ-circuitsasmodelsofeachunit cell. Aswesawintheprevious
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